#8139630

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4253 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,5,6)(3,9,11,7)(4,8,12,13)(10,15,16,14)(17,18)(19,26)(20,24)(21,31)(22,32)(23,25)(27,28)(29,30)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(2,6)(7,9)(8,13)(14,15)(19,20)(24,26)(27,29)(28,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,23,20)(18,24,25,26)(21,27,32,29)(22,28,31,30)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,19)(18,24,25,26)(21,27,32,29)(22,30,31,28)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21)(18,22)(19,27)(20,29)(23,32)(24,28)(25,31)(26,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8046826);